The present invention relates to a method of deposition profile simulation, and particularly to a simulation method to be applied for estimating microscopic profile of an axis-symmetrical structure, such as a contact hole, provided on VLSI (Very Large Scale Integrated) circuits after deposition processing, with a calculation time comparatively short making use of analytical integration.
There have been proposed several simulation methods for estimating profile evolution of trenches and contact holes in CVD (Chemical Vapor Deposition) processes. Examples of them are described in a paper entitled "A 3-dimensional model for low-pressure chemical-vapor-deposition step coverage in trenches and circular vias", by M. M. IslamRaja et al., pp. 7137 to 7140, Journal of Applied Physics 70 (11), December 1991, and a paper entitled "Influence of surface-activated reaction kinetics on low-pressure chemical vapor deposition conformality over micro features", by J. J. Hsieh, pp. 78 to 86, Journal of Vacuum Science and Technology A 11(1), January/February 1993.
In these papers, following assumptions are applied, considering that dimension of a mean-free path of a molecule in the rarefied species gas is sufficiently larger than dimension of the contact hole, for example, since inter-collision probability of the gas molecules is very small in the condition generally applied for metal CVD processes, and so, the gas molecules can be treated to arrive straight on the substrate surface from a region in a mean-free path length around the substrate surface.
1. There is but negligible inter-collisions of the gas molecules in the region of interest.
2. Molecules in the gas phase act as those of an ideal gas having a uniform molecule density with their speed following the Maxwellian distribution, and the internal freedom degree of the molecules can be neglected.
3. Directional distribution of the molecules re-emitted at the substrate surface is isotropic, since the molecules attain a thermal equilibrium on the substrate surface, losing information of their incident angles.
4. Diffusion from the substrate surface is negligible.
5. Value of the reactive sticking coefficient (hereafter abbreviated as Sc) is independent of location, that is, independent of flux density or surface conditions.
6. Sc of the incident flux is deposited and all of the other, that is (1-Sc) of the incident flux is re-emitted.
In the simulation methods of deposition profile described in the above papers, the deposition speed is obtained by calculating incident flux densities, regarding the deposition speed at a surface position to be proportional to the incident flux density at the surface position. Sc of the incident flux at a surface position, which is a sum of flux from the rarefied species gas and the flux re-emitted at the other surface positions, being deposited on the surface position, and the other of the incident flux being reflected to the other surface positions, the profile evolution at each surface position is calculated so that material income and expenses corresponding to the incident flux and the re-emitted flux should balance at each surface position in each time step.
However, as for the calculation method for obtaining value of the incident flux, there is no concrete description in the paper of M. M. IslamRaja et al., and in the paper of J. J. Hseih, there is described but a method applicable only for deposition profile of the trenches.
As for "shape factors" which describe the incident flux re-emitted from the other surface positions, they are represented by a following equation in the paper of M. M. IslamRaja et al., which can not be applied, however, directly to the trenches or the contact holes. EQU F.sub.ji=cos .psi..sub.j cos .psi..sub.i dA.sub.j dA.sub.i /.pi.r.sup.2.sub.ij
where;
F.sub.ji are shape factors describing the probability that a molecule emitted from a differential area element dA.sub.j at a surface position r.sub.j (from a suitable point of origin) strikes a region within the differential area dA.sub.i at another surface position r.sub.i,
cos .psi. represents the directional cosine, and
r.sub.ij is the magnitude of the vector represented by r.sub.j -r.sub.i.
In the paper of J. J. Hsieh, on the other hand, they are represented by a following equation, which is applicable, however, only to the trenches and furthermore, expressed only with visible angles S.sub.i,j inconvenient to be applied in actual calculation. ##EQU1## where;
.OMEGA..sub.i,j is the jth edge visibility on the ith edge,
.omega..sub.i is the surface normal angle on the ithe edge of the 2-dimensional cross section of a micro feature, and
.DELTA..sub.i,j is the Dirac Delta function.
In summary, there are disclosed fundamental equations concerning to the trenches and contact holes for defining the incident flux from gas phase and other surface positions in the prior arts, but they are not concrete in the paper of M. M. IslamRaja et al., or not convenient to be applied to the string model analysis, in the paper of J. J. Hsieh.
However, it is indispensable to calculate and integrate the flux density incident to each differential section of a 3-dimensional string model for estimating the deposition profile, and computational time depends deeply on the integration method.